Eindhoven University of Technology MASTER Tensor network methods for quantum simulation van Eersel, H. Award date: 2011 Link to publication Disclaimer This document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Student theses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the document as presented in the repository. The required complexity or quality of research of student theses may vary by program, and the required minimum study period may vary in duration. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain Tensor network methods for quantum simulation Harm van Eersel December, 2010 Abstract Quantum computing is a new paradigm in computer science, based on the phenomena of quantum physics. To classically represent an arbitrary quantum state, an exponential number of classical resources is necessary. Tensor network methods { based on a graph theoretic interpretation of ten- sors { do allow for an efficient classical representation of a limited class of quantum states. In this thesis several tensor network methods are reviewed, including matrix product states, tree tensor networks and projected entangled pair states their construction, transformation and measurement. Furthermore, some applications are discussed briefly. Also, some original work is presented for projected entangled pair states: an update procedure for unitary quantum operators, a more efficient construction method in the teleportation picture and a comparison of the different geometries. Contents Introduction 3 1 Preliminaries 5 1.1 Linear algebra . .5 1.1.1 Vector spaces . .5 1.1.2 Hilbert spaces . .7 1.1.3 Linear operators . .7 1.1.4 Singular value decomposition . .9 1.2 Tensor algebra . .9 1.2.1 Tensor product spaces . .9 1.2.2 Index notation . 10 1.2.3 Tensor contraction . 10 1.2.4 Tensor networks . 11 1.3 Quantum computing . 12 1.3.1 Quantum states . 13 1.3.2 Quantum operations . 14 1.3.3 Quantum teleportation . 16 1.3.4 Schmidt decomposition . 17 1.4 Spin systems . 17 1.4.1 Properties of spin systems . 17 1.4.2 Bosons and fermions . 18 1.4.3 Example of a spin system . 18 2 Matrix product states and tree tensor networks 20 2.1 Matrix product states . 20 2.1.1 Structure . 20 2.1.2 Construction . 21 2.1.3 Quantum operations . 22 2.1.4 Matrix product states as tensor networks . 23 2.2 Tree tensor networks . 24 2.2.1 Structure . 24 2.2.2 Construction . 25 2.2.3 Quantum operations . 25 2.2.4 Relation with MPS . 26 2.2.5 Enhancements . 27 3 Projected entangled pair states 28 3.1 Basics . 28 3.1.1 Structure . 28 3.1.2 Construction . 30 3.1.3 Quantum operations . 31 3.1.4 Extensions . 34 1 3.2 The teleportation picture . 35 3.2.1 Construction . 35 3.2.2 Quantum operations . 36 3.2.3 A more efficient variant . 37 3.3 Complexity of PEPS . 39 3.3.1 2D PEPS as cluster state . 39 3.3.2 A sharp bound . 39 3.4 Capacity of PEPS . 40 4 Other tensor network methods 43 4.1 Contracting Tensor Network formalism . 43 4.1.1 Structure . 43 4.1.2 Contraction . 44 4.1.3 Measurement . 44 4.2 Enhanced methods . 45 4.2.1 Multi-scale entanglement renormalization ansatz . 45 4.2.2 String-Bond states . 46 4.2.3 Entangled-plaquette states . 46 4.2.4 Correlator product states . 46 4.2.5 Weighted graph states . 47 4.2.6 Renormalization algorithm with graph enhancement . 47 4.2.7 Sequentially generated states . 47 4.2.8 Concatenated tensor network states . 47 4.2.9 Complete-graph tensor network states . 47 4.3 Other methods . 48 4.3.1 Matchgates . 48 4.3.2 Stabilizer formalism . 48 5 Applications 49 5.1 Physics . 49 5.1.1 Variational methods . 49 5.1.2 Renormalization group . 49 5.1.3 Density matrix renormalization group . 50 5.2 Quantum chemistry . 52 5.2.1 Encoding molecules as tensor networks . 52 6 Discussion and conclusion 53 Acknowledgements 54 2 Introduction Quantum computing is a new computing paradigm based on quantum physics instead of classical physics. Using the phenomena of quantum physics for computation, it is believed that a larger class of problems can be solved more efficiently than on classical computers. Quantum physics introduces the new concept of the quantum state. Compared with a classical state, the number of coefficients needed for the description of a state grows exponentially in the number of quantum bits, resulting in an exponential classical representation of these states. While this larger state space is essential for the additional power of quantum computers, it also makes the simulation of quantum systems intractable on classical computers. It turns out, however, that some classes of quantum systems do allow for an efficient classical simulation. Several representation methods have been proposed in the literature. They differ in the classes of states that can be represented efficiently and the operations that can be applied to these states. An ideal representation method has the following properties: • polynomial space complexity in the number of quantum bits for an interesting class of quan- tum states • a construction method to represent an arbitrary quantum state • an efficient update procedure of the representation after quantum operations on the state • efficient (local) measurement of the state and an efficient update procedure to represent the post-measurement state A number of the proposed formalisms that fulfill (some) of these requirements are based on so-called tensor networks (TN). Tensors are a generalization of the familiar concepts of scalars, vectors and matrices. Tensor networks are a graph-theoretic interpretation of tensors, expressing them in terms of vertices and edges. It has turned out that the class of quantum systems that can be simulated efficiently by TN methods are also of interest in the field of solid state physics, for example the simulation of spin systems. This allows physicists to determine relevant properties with unprecedented precision. Only recently (April 2009), it was discovered that these methods could also be applied to two- dimensional fermionic systems, enlarging the simulatable class to include many systems of chemical interest. These systems were previously thought to be unsimulatable because of the so-called sign problem associated with fermions. This has sparked considerable interest for TN methods in the field of quantum chemistry. In the field of computer science, tensor network methods are of interest because they can be used for the simulation of quantum computations. Quantum computations that can be simulated efficiently on a classical computer are in a sense not genuinely quantum. Sharpening the boundary between classes of computations that can and cannot be simulated efficiently, helps us to determine the genuine quantum elements. There are some review articles available on tensor network methods [PVWC06, VMC08] and even a recent book on quantum simulation in general [VMH09]. These works either do not incorpo- rate all tensor network based methods, or do not treat them from a computer science perspective. This thesis tries to provide an overview of the tensor network methods from a computer science perspective. 3 The thesis is organized as follows: after the preliminaries, we will first look at two closely related methods, namely matrix product states and tree tensor networks. The second chapter is the main topic of the thesis: projected entangled pair states. The chapter contains original work on update procedures, construction methods and on the question which classes of states can be represented efficiently. Chapter 3 contains an overview of other tensor network methods, including recent developments. Finally, we will look at some applications of tensor network methods in the field of solid states physics and quantum chemistry. This includes a description of the density-matrix renormalization group method. The reader is assumed to have some background in (theoretical) computer science, graph theory and linear algebra. The preliminaries of tensor calculus, quantum computing and spin systems will be discussed briefly in the next chapter. Nevertheless, some background in quantum physics and chemistry, or quantum computing and information science will certainly help. 4 Chapter 1 Preliminaries In this chapter several concepts will be introduced that lay the grounds for the rest of the thesis. We will start with the basics of linear algebra and tensor algebra, which will be used in the discussion of quantum mechanics and quantum computing that follows. The presentation is partly based on [NC00]. The chapter will be closed with an introduction to spin systems, physical models for which tensor network methods have proven invaluable. Since tensor network methods were initially developed for this type of simulation, some of the characteristics can be traced back to these systems. For a more thorough introduction to linear and tensor algebra the reader is referred to the book of Isham [Ish89] and for quantum computing to the book of Nielsen and Chuang [NC00] or, for a more concise version, to the paper of Rieffel and Polak [RP00]. 1.1 Linear algebra Linear algebra can be seen, amongst others, as the mathematical language of quantum mechanics. The field of linear algebra studies vectors spaces and linear operations on vector spaces.